Question: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-x-2y &= -7 \\ 7x+2y &= -7\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $7x = -2y-7$ Divide both sides by $7$ to isolate $x$ $x = {-\dfrac{2}{7}y - 1}$ Substitute this expression for $x$ in the first equation. $-({-\dfrac{2}{7}y - 1}) - 2y = -7$ $\dfrac{2}{7}y + 1 - 2y = -7$ Simplify by combining terms, then solve for $y$ $-\dfrac{12}{7}y + 1 = -7$ $-\dfrac{12}{7}y = -8$ $y = \dfrac{14}{3}$ Substitute $\dfrac{14}{3}$ for $y$ in the top equation. $-x-2( \dfrac{14}{3}) = -7$ $-x-\dfrac{28}{3} = -7$ $-x = \dfrac{7}{3}$ $x = -\dfrac{7}{3}$ The solution is $\enspace x = -\dfrac{7}{3}, \enspace y = \dfrac{14}{3}$.